Setting the correct Python data_path for vertex motion?
Handler workaround.

Personally would  lean towards using shape-keys for this. 

Not sure why the animated mesh doesn't update, hopefully someone will answer with a required setting, or it is an oversight in deps graph and may change up in future.

As a work-around, calling mesh.update() on a frame change via a handler gives the desired result

import bpy

def animate_meshes(scene):
    meshes = set(o.data for o in scene.objects if o.type == 'MESH')
    for m in meshes:
        m.update()

bpy.app.handlers.frame_change_post.append(animate_meshes)


clear all frame_change post handlers with

bpy.app.handlers.frame_change_post.clear()